Hello Swati.Swati said:1.Construct a matrix whose null space consists of all linear combination of the vectors, v1={1;-1;3;2} and v2={2,0,-2,4} (v1,v2 are column vector).2.The equation x1+x2+x3=1 can be viewed as a linear system of one equation in three unknowns. Express its general solution as a particular solution plus the general solution of the corresponding homogeneous system.
Row space refers to the set of all possible linear combinations of the rows of a matrix, while column space refers to the set of all possible linear combinations of the columns. Null space, also known as the kernel, is the set of all vectors that when multiplied by the matrix result in a zero vector.
Row space and column space are related through the rank of the matrix. The rank is the number of linearly independent rows or columns in the matrix. The dimension of the row space is equal to the rank, and the dimension of the column space is also equal to the rank.
To find the row space and column space of a matrix, you can use row reduction techniques to put the matrix into echelon form. The non-zero rows in the echelon form will form the basis for the row space, while the pivot columns will form the basis for the column space. To find the null space, you can solve the homogeneous system of equations using Gaussian elimination and the solutions will form the basis for the null space.
Row space, column space, and null space are important concepts in linear algebra as they provide a way to understand the properties and behavior of matrices. The row space and column space can help determine the rank and invertibility of a matrix, while the null space can be used to find solutions to homogeneous systems of equations.
Yes, it is possible for the row space and column space of a matrix to be equal. This occurs when the matrix is a square matrix, and the rank is equal to the number of rows or columns. In this case, the basis for the row space and column space will consist of the same vectors, and the dimensions of the row and column space will also be equal.